Multilateration is a method for tracking objects, i.e., for retrieving reliable information about the state of one or more objects. The state might comprise the position and/or the velocity of the objects, but is not restricted to these quantities. In general, the state can comprise all information about the object that is considered necessary, for example, the acceleration of the objects. There is a wide range of applications for multilateration methods. Particularly relevant are wide-area multilateration (WAM) methods employed in civil or military air surveillance, i.e., for tracking the positions, velocities, etc. of aircraft or other objects like, e.g., helicopters, unmanned aerial vehicles, or satellites. Another example for multilateration is the localisation of a cell phone in case of an emergency call. Multilateration methods can also be used for automotive applications based on retrieving information about the objects in the environment of a car, for example, to increase the security or the driving comfort.
The basic principle of multilateration is to receive signals transmitted by one or more objects to be tracked at several receivers, in the following also designated as sensors, located at different receiver sites, in the following also designated as sensor positions, and to make use of the different receiving times of each of the signals measured at the different receiver sites. To that end, a multilateration system comprises at least a number of receivers and a processing unit configured to process the receiving times of a signal measured at each of the receivers and to output an estimated set of information comprising, e.g., the respective positions of the one or more objects.
Various assemblies and methods can be deployed for surveillance. For example, primary or secondary radar can be used. Primary radar uses the reflection of a signal at the surface of an object. Thus, it can be used in order to localise “uncooperative” objects that are not autonomously emitting signals and/or do not answer requests solicited by the multilateration system. In a secondary radar system, signals transmitted from the multilateration system by means of an “interrogator” are answered by a transponder of an object. Such a system is also called an “active system”. In contrast to this, a “passive system” makes use of signals autonomously transmitted by an object or solicited by other equipment. Furthermore, it can be distinguished between deterministic and stochastic multilateration methods.
A rather straightforward technique for finding the position of an object by means of the above described equipment is a deterministic method as follows.
A relation between an object's position emitting a signal and the receiving times of the signal at each of the receivers is given by a set of N equations
                                          y            i                    =                                                                                                          x                    _                                    -                                                            S                      _                                        i                                                                              c                        +                                          t                e                            ⁡                              (                                  i                  ∈                                      {                                          1                      ,                      …                      ⁢                                                                                          ,                      N                                        }                                                  )                                                    ,                            (        1        )            with x denoting the position of the object and Si denoting the position of the i'th receiver, N labelling the number of receivers, c denoting the propagation velocity of the medium (ca. 3×108 m/s in air), te labelling the emission time of the signal, yi being the measured receiving times at the i'th receiver, and ∥·∥ denoting the Euclidean norm or L2 norm in space. Usually, the dimension D of the space, and thus, also the dimension of the vectors x and Si, is 2 or 3. In order to remove the unknown emission time te, the equation set (1) is transformed into another equation set comprising the time differences of arrival (TDOA) instead of the measured receiving times. Therefore, without loss of generality a reference receiver, for example the receiver at S1, may be chosen and the measured receiving times of each of the other receivers may be subtracted from the measured receiving time at S1. The result is a set of (N−1) equations, each of which describing a hyperboloid in space. The position of the object is then given by the intersection of these hyperboloids. For the 2-dimensional case, it follows that at least N=3 receivers are required for a unique determination of an object's position. Then, the position of the object is given by the intersection of 2 hyperbolas. In the 3-dimensional case, 4 receivers are necessary to locate the object at the intersection point of 3 hyperboloids. Additional receivers can be used to improve the accuracy of the result.
As the described method is a purely deterministic model, it ignores that measurements in real world are superimposed with noise. To address this issue, estimation methods have to be employed that allow for an explicit consideration of noise or measurement errors within the calculation. The issue has been discussed, for example, in Ref [1], where an “equation error” is introduced and a least-square procedure is used for an estimation of the position of an object. Under certain assumptions, the algorithm proposed in Ref [1] can be considered as a maximum-likelihood estimator. A further development is the maximum-likelihood estimator proposed in Ref [2], where additionally an intermediate variable is used. This leads to a hyperspace solution.
Furthermore, variants of the Kálmán filter proved to be especially suitable to address the problem of tracking objects. The Kálmán filter allows for an estimation of a system's state in real time, even in those cases where only information from inaccurate observations is available. When using Kálmán filters, it is prerequisite that the dynamics of the system state is described separately from the measurement process. This is normally done by (i) a system equation describing the time evolution of the system state and (ii) a measurement equation coupling the system state with the measurements. Usually, both of the equations comprise a deterministic and a stochastic part. This way, it is not only possible to describe noisy observations, but also to deal with uncertainties in modelling the dynamic behaviour of the state [10].
In a time-discretised approach, system equation and measurement equation can then be written in the form:zk+1=a(zk,wk),  (2)yk=h(zk,vk),  (3)wherein the function a recursively describes the time evolution of the system state zk by propagating it from a given time step k to the next time step (k+1), and the function h establishes for any time step k a relation for the system state zk and the measurement yk. Further, wk and vk are random variables describing uncertainties of the time evolution model of Eq. (2) or noise during the measurement process described by Eq. (3). Then, due to the stochastic nature of the Equations (2) and (3), also the state zk and the measurement yk are random variables. The time propagation of a probability distribution of the state zk can then be described by a two-step procedure comprising a prediction step and a filter step. Be fe(zk) an estimated distribution of zk at time step k. Then, in a first step, a prediction for the distribution fp(zk) of the state zk at time step (k+1) is given by the Chapman-Kolmogorov equationfp(zk+1)=∫f(zk+1|zk)·fe(zk)dzk,  (4)wherein the transition density f(zk+1|zk) is defined by Eq. (2). In the second step, the current measurement ŷk is used to filter the result of Eq. (4) according to Bayes' rulefe(zk)=ck·f(ŷk|zk)·fp(zk),  (5)where ck=1/(∫f(ŷk|zk)·fp(zk)·dzk) is a normalization constant and f(ŷk|zk) is the likelihood defined by Eq. (3).
Under the conditions that Equations (2) and (3) both are linear and that the state is normally distributed, the density fp predicted in Eq. (2) as well as the estimated density fe of the filter step in Eq. (5) can be derived exactly within the Kálmán framework that yields the first two moments, i.e., expectation value and covariance, of these distributions. As under said conditions both densities stay normally distributed for all time steps, the densities are completely described therewith.
However, the Kálmán filter method described above is restricted to purely linear models. As multilateration methods are in general based on non-linear measurement equations, the original formulation of the Kálmán filter cannot be applied. However, variants such as the extended Kálmán filter (EKF) or the unscented Kálmán filter (UKF) are available that provide approximations in case of non-linear equations. An analysis of the performance of these filters in the context of multilateration is given in Ref. [3], which also addresses the recent trend “towards the use of a number of lower cost, low fidelity sensors” [3]. The EKF has been derived from the original Kálmán filter by means of a successive linearization of the process [9]. On the other hand, the UKF or the so-called Gaussian filters are examples of sample-based linear regression Kálmán filters (LRKFs) [10]. Several types of Gaussian filters have been discussed in detail in Ref [9], and it has been shown that these filters are numerically superior over the EKF without causing additional numerical costs. In some cases, for example if the considered equations can be separated into a linear and a nonlinear substructure or if the state vector comprises a directly observed and an indirectly observed part, only part of the filtering process has to be treated in an approximate fashion, when appropriate decomposition methods are used as, for example, proposed in Ref. [10] for the class of Gaussian filters.
Attention should be paid to the fact that all multilateration methods mentioned so far, i.e., the deterministic method of Eq. (1) as well as the stochastic methods described in Refs. [1-3], do not work on the raw data obtained from the measured receiving times, but on the TDOA (time difference of arrival) measurements. This implies that these methods are ultimately based on finding the intersection point of a number of hyperboloids. As mentioned above, the reason for this is to eliminate the unknown emission time of the signals from the description. However, intersecting hyperboloids “is considered a hard task” [6]. Therefore, in Refs. [5] and [6] a multilateration method is developed that allows to reduce the problem to finding the intersection of cones, which is numerically much easier to handle. The emission time b of a signal is thereby estimated. In Ref. [5], the estimation is performed by minimising the Mahalanobis distance M(b) with respect to b.
As already outlined above, closed-form solutions as proposed in Refs. [1,2] or state estimators as disclosed in Ref. [3] can be used, when the arrival times are converted to time difference of arrival. On the other hand, if the arrival times are processed directly, the problem is equal to the GPS problem and the closed-form solution of Ref [4] can be used. Further approaches that process arrival times directly by using a state estimator can be found in Ref [5] or [6] (see above), where in the latter one a system model describes the evolution of the emission time over time. Note, that in Ref. [5], no system model is used to describe the emission time. The emission time is determined by using a certain distance measure in a separate step.
A drawback of state of the art multilateration methods and systems is that all sensors of the multilateration system need to be accurately synchronized, so that the measured receiving times can be performed with a sufficiently high precision. Thus, it is required that all sensors have the same time base so that clock offsets of each of the sensor clocks are eliminated. This synchronization requires high effort and is usually performed by means of additional reference transponders or atomic clocks. The process for determining clock offsets, so that all sensor clocks are synchronized, is called calibration.